 
\lastversion = 4
\hbadness = 10000
\hfuzz = 20 in
\title {Math 323 \qquad Test 2}
\date {March 2,1994}
\noprofessor

%\answersheetfootline {I have observed the honor code
%\hskip 10pt 
%   \underbar{\hskip 3 in}}
%version(1 green),(2 yellow),(3 blue),(4 pink)

%%%begin

%begin 1

\problem  A random variable Y is Poisson with mean 
$\lambda = 2$.  Find the expected value $E(X^2)$.
\correct $E(X^2) = 6$
\wrong $E(X^2)= 4$
\wrong $E(X^2)= 2$
\wrong $E(X^2)= 8$ 
\wrong $E(X^2) = 12$ 
\perm1:abcde
\perm2:acebd
\perm3:cbaed
\perm4:cadbe

%end 1

%begin 2  

\problem  Before accepting a shipment of  30 appliances 
a large buyer will take a sample of 4 of the 30
 and check them carefully for defects.  If more than 
one in the sample are defective the shipment
 is rejected. Of the 30, it is known that 2 are defective.
 What is the probability that the shipment will be accepted?  
\correct ${858\over 870}$
\wrong ${208\over 870}$
\wrong ${650\over 870}$
\wrong ${220\over 870}$
\wrong ${836\over 870}$
\perm1:bcaed
\perm2:acebd
\perm3:dbcea
\perm4:cadbe

%end 2



%begin 3

\problem Incoming calls at a local 
switchboard follow a Poisson distribution
with an average rate of 3 calls per minute.  
If a call comes in at 1:00, what is the
probability that the second call comes in by 1:01?

\correct .95
\wrong  .72
\wrong .83
\wrong .05
\wrong .28
\perm1:cbdae
\perm2:bdaec
\perm3:daecb
\perm4:aecbd

%end 3

%begin 4

\problem The amount of raw sugar which can be processed 
in a factory  in one day has an exponential distribution
 with a mean of 5 (tons).What is the probability
 of processing at least 6 tons in one day? 
\correct $.30$
\wrong $.42$
\wrong $.36$
\wrong $.24$
\wrong $.45$
\perm1:acbed
\perm2:cbeda
\perm3:bedac
\perm4:adbec

\pagebreak
%begin 5

\problem In problem 4, what is the probability 
that  the factory processes
at least 5 tons next week on exactly two of the first three 
days?
\correct $.257$
\wrong $.225$
\wrong $.375$
\wrong $.667$
\wrong $.312$
\perm1:baced
\perm2:acedb
\perm3:cedba
\perm4:edbac

%end5

%begin 6


\problem (Refer to problem 4 again.) How many tons of raw sugar should be 
stocked daily at the factory  so that the 
probability of running out of sugar that 
day will be less than .05?
(The factory cannot buy fractions of a 
ton, so round off your answer to the 
next integer.)
\correct $15$
\wrong $12$
\wrong $8$
\wrong $8$
\wrong $21$
\perm1:cbade
\perm2:badec
\perm3:adecb
\perm4:decba


%end 6




%begin 7

\problem A jeweller has received a shipment of 100 diamonds.
He knows that 20\% of these are of top quality.  
What is the probability 
that 50 diamonds must be examined to get 10 of top quality?
\correct $\pmatrix{49\cr9\cr}(.2)^{10}(.8)^{40}$
\wrong $\pmatrix{50\cr10\cr}(.2)^{10}(.8)^{40}$
\wrong $\pmatrix{49\cr10\cr}(.2)^{10}(.8)^{40}$          
\wrong $\pmatrix{49\cr9\cr}(.2)^9(.8)^{40}$
\wrong $\pmatrix{50\cr10\cr}(.2)^9(.8)^{40}$
\perm1:bdaec
\perm2:daecb
\perm3:aecbd
\perm4:ecbda
%end 7


%begin 8
\problem The probability generating function of the random 
variable X is $$ P(t) = {1\over32}[1 + 5t + 10t^2 + 10t^3 + 5t^4 + t^5].$$
What is the expected value E[X(X-1)]?
\correct 5
\wrong 20
\wrong 160
\wrong 4
\wrong 6
\perm1:edcba
\perm2:daecb
\perm3:aecbd
\perm4:ecbda
%end 8

\pagebreak

%begin9

\problem The number of trucks crossing the 
stateline on I-65 southbound follows a Poisson distribution.
If the average number of crossings is 60 per hour,
what is the probability that at least one truck crosses in the next minute?

\correct $.632$
\wrong $.368$
\wrong $.721$
\wrong $.435$
\wrong $.596$
\perm1:abcde
\perm2:bcdea
\perm3:cdeab
\perm4:deabc
\perm4:eabcd
%end9

%begin10
\problem In problem 9, what is the probability that 3 
trucks cross the line 
between 12:00 to 12:02 and 2 trucks cross between 1:00
 and 1:02? 
\correct .049
\wrong .451
\wrong .156
\wrong .206
\wrong .317
\perm1:bacde
\perm2:acdeb
\perm3:cdeba
\perm4:debac
%end 10



%begin 11
\problem An archer manages to hit the bulls-eye of his
target 30\% of the time.  What is his probability of hitting the bulls-eye
at least 2 times in 10 tries?
\correct .851
\wrong .912
\wrong.741
\wrong .776
\wrong .827
\perm1:baecd
\perm2:aecdb
\perm3:ecdba
\perm4:cdbae
%end 11


%begin 12

\problem Let X be a number chosen at
 random from the  real numbers in the interval (5,10). 
What is the probability that 3 
times the distance from X to 5 is greater than 4 times the distance
from X to 10?
\correct ${3\over 7}$
\wrong ${6\over7}$
\wrong ${4\over7}$
\wrong ${5\over7}$
\wrong ${2\over7}$
\perm1:ecbda
\perm2:bdaec
\perm3:dabce
\perm4:dabce

%end12

\pagebreak

%begin 13
\problem What does the sum $$\sum_{y=1}^\infty {{y^2}{{(1-p)}^{y-1}}{p}}$$ equal?
\correct ${{2-p}\over {p^2}}$
\wrong ${{1-p}\over {p^2}}$
\wrong ${1\over {p^2}}$
\wrong ${{1-p}\over p}$
\wrong ${p\over {{(1-p)}^2}}$
\perm1:baecd
\perm2:cdbae
\perm3:dbaec
\perm4:ecdba

%end 13


%begin 14
\problem What does the integral $$\int_0^\infty{1\over 9}{x^2}{e^{-{x\over 3}}}dx$$
equal?
\correct 6
\wrong 18
\wrong 9
\wrong 27
\wrong 12
\perm1:ecbad
\perm2:decba
\perm3:ecbad
\perm4:cbade

%end 14

%begin 15
\problem For which value of k is the function, 
$$f(x)=\left\{ 
\matrix{
$k{x^5}$ & \hbox{for}$\, 0\,\leq x\,\leq 1$ \cr 
$0$ &  \hbox{elsewhere}\cr
}
\right.,
$$ a probability density function?
         
\correct 6
\wrong ${1\over 6}$
\wrong ${1\over 5}$
\wrong 5
\wrong 7
\perm1:cbeda
\perm2:edacb
\perm3:dacbe
\perm4:cbeda

%end 15



%begin 16


\problem Which one of the following 5 functions cannot be a probability density 
function, no matter what value of k we choose?  All 5 are defined to be 0 for x less than 1,
but for x greater than 1 are defined by
\correct ${k\over x}$
\wrong ${k\over{x^2}}$
\wrong $k{e^{-x}}$
\wrong ${kx{e^{-x}}}$
\wrong ${k\over{x^3}}$
\perm1:cdeab
\perm2:cdeab
\perm3:deabc
\perm4:abcde

%end16
\pagebreak

%begin 17
\problem  A pair of dice is rolled until a sum of 7 appears.  What is the probabilty
that this occurs on the third roll?
\correct .116
\wrong .342
\wrong .193
\wrong .218
\wrong .095
\perm1:adbec
\perm2:becad
\perm3:becad
\perm4:ecadb

%end17


%begin 18
\problem  The number of radioactive particles emitted from a radioactive substance
follows a Poisson distribution with a mean of 2 particles a minute.
What is the average time elapsed for 5 particles to be emitted?
\correct  2.5 minutes
\wrong 10 minutes
\wrong 20 minutes
\wrong 5 minutes
\wrong 1.25 minutes
\perm1:edacb
\perm2:edacb
\perm3:dacbe
\perm4:edacb

%end18



\end